Average Error: 1.0 → 1.0
Time: 25.4s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2
double f(double g, double h) {
        double r6025836 = 2.0;
        double r6025837 = atan2(1.0, 0.0);
        double r6025838 = r6025836 * r6025837;
        double r6025839 = 3.0;
        double r6025840 = r6025838 / r6025839;
        double r6025841 = g;
        double r6025842 = -r6025841;
        double r6025843 = h;
        double r6025844 = r6025842 / r6025843;
        double r6025845 = acos(r6025844);
        double r6025846 = r6025845 / r6025839;
        double r6025847 = r6025840 + r6025846;
        double r6025848 = cos(r6025847);
        double r6025849 = r6025836 * r6025848;
        return r6025849;
}
↓
double f(double g, double h) {
        double r6025850 = 0.6666666666666666;
        double r6025851 = atan2(1.0, 0.0);
        double r6025852 = g;
        double r6025853 = -r6025852;
        double r6025854 = h;
        double r6025855 = r6025853 / r6025854;
        double r6025856 = acos(r6025855);
        double r6025857 = 3.0;
        double r6025858 = r6025856 / r6025857;
        double r6025859 = fma(r6025850, r6025851, r6025858);
        double r6025860 = cos(r6025859);
        double r6025861 = 2.0;
        double r6025862 = r6025860 * r6025861;
        return r6025862;
}

Error

Bits error versus g

Bits error versus h

Derivation

  1. Initial program 1.0

    \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]

  2. Simplified1.0

    \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2}\]

  3. Final simplification1.0

    \[\leadsto \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))